Question: Find one value of $x$ that is a solution to the equation: $(x-2)^2-6(x-2)+5=0$ $x=$
Answer: We could solve for $x$ by expanding $(x-2)^2$ and $-6(x-2)$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that if we let ${p}={x-2}$, we can rewrite the equation: $({x-2})^2-6({x-2})+5=0$ In particular, we can express it in the form: ${p}^2-6{p}+5=0$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2-6{p}+5&=0\\\\ ({p}-5)({p}-1)&=0\\\\ {p}=5\ &\text{or} \ \ {p}=1 \end{aligned}$ Since ${p}={x-2}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${x-2}=5\ \ \ \text{or} \ \ \ {x-2}=1$ When we solve $x-2=5$, we find that $x=7$. When we solve $x-2=1$, we find that $x=3$. In conclusion, the two solutions of the equation $(x-2)^2-6(x-2)+5=0$ are $x=7$ and $x=3$. [Is there another way to solve for x?]